Journal of Applied Probability

Asymptotic properties of a leader election algorithm

Ravi Kalpathy, Hosam M. Mahmoud, and Mark Daniel Ward

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

Article information

Source
J. Appl. Probab., Volume 48, Number 2 (2011), 569-575.

Dates
First available in Project Euclid: 21 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1308662645

Digital Object Identifier
doi:10.1239/jap/1308662645

Mathematical Reviews number (MathSciNet)
MR2840317

Zentralblatt MATH identifier
1219.60008

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees 60F05: Central limit and other weak theorems 68W40: Analysis of algorithms [See also 68Q25]

Keywords
Leader election analytic combinatorics Rice's method fixed point contraction method Wasserstein metric space weak convergence

Citation

Kalpathy, Ravi; Mahmoud, Hosam M.; Ward, Mark Daniel. Asymptotic properties of a leader election algorithm. J. Appl. Probab. 48 (2011), no. 2, 569--575. doi:10.1239/jap/1308662645. https://projecteuclid.org/euclid.jap/1308662645


Export citation

References

  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Clarendon Press, Oxford.
  • Devroye, L. (1992). A limit theory for random skip lists. Ann. Appl. Prob. 2, 597–609.
  • Fill, J. A., Mahmoud, H. M. and Szpankowski, W. (1996). On the distribution for the duration of a randomized leader election algorithm. Ann. Appl. Prob. 6, 1260–1283.
  • Flajolet, P. and Sedgewick, R. (1995). Mellin transforms and asymptotics: finite differences and Rice's integrals. Theoret. Comput. Sci. 144, 101–124.
  • Janson, S. and Szpankowski, W. (1997). Analysis of an asymmetric leader election algorithm. Electron. J. Combinatorics 4, R17, 16pp.
  • Karr, A. F. (1993). Probability. Springer, New York.
  • Louchard, G. and Prodinger, H. (2008). Advancing in the presence of a demon. Math. Slovaca 58, 263–276.
  • Louchard, G., Prodinger, H. and Ward, M. D. (2011). Number of survivors in the presence of a demon. To appear in Periodica Mathematica Hungarica.
  • Neininger, R. (2005). Recursive random variables with subgaussian distributions. Statist. Decisions 23, 131–146.
  • Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Prob. 14, 378–418.
  • Papadakis, T., Munroe, J. I. and Poblete, P. V. (1990). Analysis of the expected search cost in skip lists. In SWAT 90 (Bergen, 1990; Lecture Notes Comput. Sci. 447), Springer, Berlin, pp. 160–172.
  • Prodinger, H. (1993). How to select a loser. Discrete Math. 120, 149–159.
  • Pugh, W. (1989). Skip lists: a probabilistic alternative to balanced trees. In Algorithms and Data Structures (Ottawa, ON, 1989; Lecture Notes Comput. Sci. 382), Springer, Berlin, pp. 437–449.
  • Rösler, U. (1991). A limit theorem for “Quicksort”. RAIRO Inf. Théor. Appl. 25, 85–100.
  • Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.